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2018 AMC 10B problems

All 25 problems from the 2018 AMC 10B, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread…Geometry
  2. 2Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second…Algebra
  3. 3Problem 3In the expression (underline phantom x × underline phantom x) + (underline phantom x × underline phantom x) each blank is to be filled in with one of…Counting & Probability
  4. 4Problem 4A three-dimensional rectangular box with dimensions X, Y, and Z has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is X…Geometry
  5. 5Problem 5How many subsets of {2, 3, 4, 5, 6, 7, 8, 9} contain at least one prime number?Counting & Probability
  6. 6Problem 6A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn…Counting & Probability
  7. 7Problem 7In the figure below, N congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the…Algebra
  8. 8Problem 8Sara makes a staircase out of toothpicks as shown: This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that…Algebra
  9. 9Problem 9The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let p be the probability that when all 7 dice are rolled, the sum of…Counting & Probability
  10. 10Problem 10In the rectangular parallelepiped shown, AB = 3, BC = 1, and CG = 2. Point M is the midpoint of FG. What is the volume of the rectangular pyramid…Geometry
  11. 11Problem 11Which of the following expressions is never a prime number when p is a prime number?Number Theory
  12. 12Problem 12Line segment AB is a diameter of a circle with AB = 24. Point C, not equal to A or B, lies on the circle. As point C moves around the circle, the…Geometry
  13. 13Problem 13How many of the first 2018 numbers in the sequence 101, 1001, 10001, 100001, … are divisible by 101?Number Theory
  14. 14Problem 14A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the…Algebra
  15. 15Problem 15A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices…Geometry
  16. 16Problem 16Let a_1, a_2, …, a_2018 be a strictly increasing sequence of positive integers such that a_1 + a_2 + … + a_2018 = 2018^2018. What is the remainder…Number Theory
  17. 17Problem 17In rectangle PQRS, PQ = 8 and QR = 6. Points A and B lie on PQ, points C and D lie on QR, points E and F lie on RS, and points G and H lie on SP so…Algebra
  18. 18Problem 18Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in…Counting & Probability
  19. 19Problem 19Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the…Number Theory
  20. 20Problem 20A function f is defined recursively by f(1) = f(2) = 1 and f(n) = f(n - 1) - f(n - 2) + n for all integers n ≥ 3. What is f(2018)?Algebra
  21. 21Problem 21Mary chose an even 4-digit number n. She wrote down all the divisors of n in increasing order from left to right: 1, 2, …, n/2, n. At some moment…Number Theory
  22. 22Problem 22Real numbers x and y are chosen independently and uniformly at random from the interval [0, 1]. Which of the following numbers is closest to the…Geometry
  23. 23Problem 23How many ordered pairs (a, b) of positive integers satisfy the equation a · b + 63 = 20 · lcm(a, b) + 12 · gcd (a, b), where gcd (a, b) denotes the…Number Theory
  24. 24Problem 24Let ABCDEF be a regular hexagon with side length 1. Denote by X, Y, and Z the midpoints of sides AB, CD, and EF, respectively. What is the area of…Geometry
  25. 25Problem 25Let ⌊ x ⌋ denote the greatest integer less than or equal to x. How many real numbers x satisfy the equation x^2 + 10,000⌊ x ⌋ = 10,000x?Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.